Semidefinite Programming for Min-Max Problems and Games

TL;DR

This paper develops a unified semidefinite programming approach to approximate solutions for min-max problems and zero-sum polynomial games, enabling efficient computation of Nash equilibria and strategies.

Contribution

It introduces a hierarchy of semidefinite relaxations for min-max problems and polynomial games, providing a practical and unified method for approximating solutions.

Findings

Hierarchies of semidefinite relaxations effectively approximate solutions.

Few relaxations often suffice for good approximations or finite convergence.

Polynomial-time solvability of each relaxation in the hierarchy.

Abstract

We introduce two min-max problems: the first problem is to minimize the supremum of finitely many rational functions over a compact basic semi-algebraic set whereas the second problem is a 2-player zero-sum polynomial game in randomized strategies and with compact basic semi-algebraic pure strategy sets. It is proved that their optimal solution can be approximated by solving a hierarchy of semidefinite relaxations, in the spirit of the moment approach developed in Lasserre. This provides a unified approach and a class of algorithms to approximate all Nash equilibria and min-max strategies of many static and dynamic games. Each semidefinite relaxation can be solved in time which is polynomial in its input size and practice from global optimization suggests that very often few relaxations are needed for a good approximation (and sometimes even finite convergence).